#416391

order	512
nilpotency class	3
isoclinism	isoclinic class of #393899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

961 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10,13,18,14,19,15,29)(11,20,22,32,24,36,23,16)(12,25,27,35,28,17,30,34)(21,37,26,40,38,33,39,31);(9,11,15,22,14,24,13,23)(10,16,29,20,19,32,18,36)(12,26,30,38,28,39,27,21)(17,33,35,31,25,37,34,40);(2,5)(6,7)(9,12)(10,17)(11,21)(13,27)(14,28)(15,30)(16,31)(18,34)(19,25)(20,37)(22,26)(23,39)(24,38)(29,35)(32,40)(33,36);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=e,b^2=efh,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323587);